On the spectrum of Riemannian submersions with totally geodesic fibers

Gérard Besson; Manlio Bordoni

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1990)

  • Volume: 1, Issue: 4, page 335-340
  • ISSN: 1120-6330

Abstract

top
In this Note we give a rule to compute explicitely the spectrum and the eigenfunctions of the total space of a Riemannian submersion with totally geodesic fibers, in terms of the spectra and eigenfunctions of the typical fiber and any associated principal bundle.

How to cite

top

Besson, Gérard, and Bordoni, Manlio. "On the spectrum of Riemannian submersions with totally geodesic fibers." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 1.4 (1990): 335-340. <http://eudml.org/doc/244285>.

@article{Besson1990,
abstract = {In this Note we give a rule to compute explicitely the spectrum and the eigenfunctions of the total space of a Riemannian submersion with totally geodesic fibers, in terms of the spectra and eigenfunctions of the typical fiber and any associated principal bundle.},
author = {Besson, Gérard, Bordoni, Manlio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Laplace-Beltrami operator; Riemannian submersion; Representations of Lie groups; spectrum; Laplace operator; totally geodesic fibers},
language = {eng},
month = {12},
number = {4},
pages = {335-340},
publisher = {Accademia Nazionale dei Lincei},
title = {On the spectrum of Riemannian submersions with totally geodesic fibers},
url = {http://eudml.org/doc/244285},
volume = {1},
year = {1990},
}

TY - JOUR
AU - Besson, Gérard
AU - Bordoni, Manlio
TI - On the spectrum of Riemannian submersions with totally geodesic fibers
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1990/12//
PB - Accademia Nazionale dei Lincei
VL - 1
IS - 4
SP - 335
EP - 340
AB - In this Note we give a rule to compute explicitely the spectrum and the eigenfunctions of the total space of a Riemannian submersion with totally geodesic fibers, in terms of the spectra and eigenfunctions of the typical fiber and any associated principal bundle.
LA - eng
KW - Laplace-Beltrami operator; Riemannian submersion; Representations of Lie groups; spectrum; Laplace operator; totally geodesic fibers
UR - http://eudml.org/doc/244285
ER -

References

top
  1. BÉRARD BERGERY, L. - BOURGUIGNON, J. P., Laplacians and Riemannian submersions with totally geodesic fibres. Illinois Journal of Math., 26, n. 2, 1982, 181-200. Zbl0483.58021MR650387
  2. BESSON, G., A Kato type inequality for Riemannian submersions with totally geodesic fibers. Annals of Global Analysis and Geometry, vol. 4, n. 3, 1986, 273-289. Zbl0631.53035MR910547DOI10.1007/BF00128049
  3. BREDON, G., Introduction to compact transformation groups. Academic Press, 1972. Zbl0246.57017MR413144
  4. CHEEGER, J., Some examples of manifolds of non negative curvature. J. Diff. Geom., 8, 1972, 623-629. Zbl0281.53040MR341334
  5. DONNELLY, H., G-spaces, the asymptotic splitting of L 2 ⁢ M into irreducibles. Math. Annalen, 237, 1978, 23-40. Zbl0379.53019MR506653DOI10.1007/BF01351556
  6. HERMANN, R., A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle. Proc. Amer. Math. Soc, vol. II, 1960, 236-242. Zbl0112.13701MR112151
  7. KOBAYASHI, S. - NOMIZU, K., Foundations of differential Geometry. Vol. I. Wiley Interscience, New York-London1963. Zbl0119.37502MR152974
  8. O'NEILL, B., The fundamental equations of a submersion. Michigan Math. J., 13, 1966, 459-469. Zbl0145.18602MR200865
  9. SERRE, J. P., Representations linéaires des groupes finis. Hermann, Paris1971. Zbl0223.20003MR352231
  10. VILMS, J., Totally geodesic maps. J. Diff. Geom., 4, 1970, 73-79. Zbl0194.52901MR262984
  11. WARNER, G., Harmonic analysis on semisimple Lie groups. Vol. I. Springer Verlag, 1972. Zbl0265.22020

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.